No superflya, in uniform circular motion there is always the centripetal force, which is directed towards the centre of the circle. The force changes the 'direction' of the speed, and hence the velocity, but it doesn't actually affect the speed.
It doesn't affect the speed because the centripetal force is at all times perpendicular to the velocity. In a sense, it 'tugs' the velocity vector to change direction, but it doesn't change the size of the vector (which is the speed).
Take special care of the word
uniform. Uniform circular motion occurs when the only force is towards the centre of the circle.
If you vertically swing a ball tied to a string , it is not undergoing uniform circular motion because it is being affected by the gravitational field, and thus it does not move at constant speed.
The constancy of speed can be proven using mathematics, but isn't that elementary if you haven't learnt about vectors.
From the unit circle in trigonometry,
,
Therefore, two equations describing the circle are
,
Now, to turn this into a dynamic system, where the object is moving around the circle, we can replace
with 't' for time.
,
The period of the motion around the circle in this case will be
(the period of sin and cos), but that's not very important.
If we differentiate both with respect to time:
These are the horizontal and vertical velocities of the object. Imagine these as being the magnitudes of vectors, pointing horizontally and vertically respectively.
The resultant sum of these vectors is the velocity,
And the length of the velocity vector is the speed!
^2+\left(\frac{dy}{dt}\right)^2} = \sqrt{(-\sin{t})^2+(\cos{t})^2} = 1=constant)
.
So the speed of an object moving around a circle is constant. No matter what radius or period we choose, the result still stands.
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